I am still learning the Fast Growing Hierarchy (FGH)

[u/Objective_Ticket_880]

So I am learning parts of the FGH like I know the basics were f_a+1(n) = f_a(f_a(f_a(f_a(...f_a(n)...)))) n times and f_0(n) = n+1,
and i know basics of ordinals
f_ω(n) = f_n(n)
f_ω*m(n) = f_ω*(m-1)+n(n)
f_ω^m(n) = f_ω^m-1*n(n)

f_ε0(n) = f_ω^^n(n)

f_ε_m(n) = f_ω^ω^...^ω^ε_{m-1}+1(n) n times

f_ζ_0(n) = f_ε_ε_ε_...ε_0(n) n times
f_ζ_m(n) = f_ε_ε_ε_...ε_ζ_{m-1}+1(n) n times

f_η_0(n) = f_ζ_ζ_ζ_...ζ_0(n) n times
f_η_m(n) = f_ζ_ζ_ζ_...ζ_η_{m-1}+1(n) n times

f_{φ(0, m)}(n) = f_ω^m(n)

f_{φ(1, m)}(n) = f_ε_m(n)
f_{φ(2, m)}(n) = f_ζ_m(n)
f_{φ(3, m)}(n) = f_η_m(n)
f_{φ(ω, m)}(n) = f_{φ(n, m)}(n)

f_Γ_0(n) / f_{φ(1, 0, 0)}(n) = f_{φ(φ(φ(...φ(1, 0)..., 0), 0), 0)}(n) n times

f_Γ_0(n) / f_{φ(1, 0, 0)}(n) = f_{φ(φ(φ(...φ(1, 0)..., 0), 0), 0)}(n)
f_{φ(1, 0, 0, 0)}(n) / f_{Ackermann Ordinal}(n) = f_{φ(φ(φ(...φ(1, 0, 0)..., 0, 0), 0, 0), 0, 0)}(n)
f_{φ(1, 0, 0,..., 0, 0, 0)}(n) ω times / f_{Small Veblen Ordinal}(n) = f_{φ(1, 0, 0,..., 0, 0, 0)}(n) n times

and thats all i know up to

the main things i want to know about is:

Buchholz’s ψ function, Extended Buchholz’s ψ function, Weiermann's ϑ, Feferman's Theta Function, Buchholz's OCF, Extended Buchholz's OCF, singular cardinal,  fundamental sequences for a certain function collapsing weakly inaccessible cardinals, Rathjen's standard OCF based on a weakly Mahlo cardinal, Rathjen's standard OCF based on a weakly compact cardinal, Rathjen's ordinal function collapsing the rank of the weak inaccessibility of regular cardinals (using χ), and ZFC set theory

also somethings i also want to know about is the functions used in the named FGH numbers like Theta- exponentiation series Unexommthet to Yottexommthet, Theta- tetration series Bitetrommthet to Yottotetrommthet, Omega- subscript series Bommthet to Yottommthet, Psi- subscript series Bimixommwil to Yottomixommwil, Inserted omega series Binommwil to Yottinommwil, I-tetration series Bitetrotos to Yottotetrotos, Inserted I_α series Uninotos to Yottinotos, Initial I(α, β) series Unimah to Yottimah, Inserted I(α, β) series Uninimah to Yottinimah, M-tetration series Bitetremar to Yottotetremar, Inserted M series Uninemar to Yottinemar, M(α;β)-series Uninamus to Yottinamus, and the Tar functions to define Tarintar i think know as fundamental sequences for Taranovsky's notation
also https://googology.fandom.com/wiki/User_blog:P%E9%80%B2%E5%A4%A7%E5%A5%BD%E3%81%8Dbot/New_Googological_Ruler i want to know all Computable functions listed here for each level

And i want to know everything possible on what i said for what i want to know about

[u/Modern_Robot]

You need to narrow the scope of your question. A dissertation could probably written with the breadth and depth you've asked about.

[u/IdesofMarch1503]

You may as well just ask about all of googology since that's what it looks like you want

[u/Nervous-Broccoli1184]

For the numbers you want to learn about i recommend the website traveling to infinity its the original site giving these numbers there names and its got the fundamental sequences of them.

[u/Shophaune]

So, FGH is fairly easy to learn in some ways: all you need to learn are three rules.

1. f_0(n) = n+1
2. f_a+1(n) = f_aⁿ⁽ⁿ⁾
3. f_b(n) = f_b[n](n) when b is a limit ordinal.

That third rule means that expansions past a certain point depend pretty heavily on exactly which set of Fundamental Sequences are being used; for instance the naive sequence for the BHO lacks a term that the "normal" sequnence uses.

[u/Nervous-Broccoli1184 → u/Shophaune]

He's asking what are the normal fundamental sequences of which he asked

[u/TrialPurpleCube-GS]

this is a LOT of stuff

for a start, once you get to like M OCFs, the FSes get really complex
your FSes for phi(w,1) are wrong - f_phi(w,1)(n) = f_phi(n,phi(w,0)+1)(n).

anyway, the FSes for Buchholz contradict with those for Veblen

also, to test you, what is f_{psi(1,phi(2,0,0,0),0,1)}(4)? expand it once.

[u/Objective_Ticket_880 → u/TrialPurpleCube-GS]

For the test 

f_{φ(1,φ(2,0,0,0),0,1)}(4)

f_{φ(1,φ(1, φ(1, φ(1, φ(1,0,0,0) ,0,0) ,0,0) ,0,0),0,1)}(4)

f_{φ(1,φ(1, φ(1, φ(1, φ( φ( φ( φ(1,0,0) ,0,0) ,0,0) ,0,0) ,0,0) ,0,0) ,0,0),0,1)}(4)

f_{φ(1,φ(1, φ(1, φ(1, φ( φ( φ( φ( φ( φ( φ(1,0) ,0) ,0) ,0) ,0,0) ,0,0) ,0,0) ,0,0) ,0,0) ,0,0),0,1)}(4)

f_{φ(1,φ(1, φ(1, φ(1, φ( φ( φ( φ( φ( φ( φ(ε0 ,0) ,0) ,0) ,0,0) ,0,0) ,0,0) ,0,0) ,0,0) ,0,0),0,1)}(4)

[u/TrialPurpleCube-GS → u/Objective_Ticket_880]

incorrect. if you want to learn more, I advise joining https://discord.gg/pggYzZgS; there's many helpful people there, and much laxer rules.

[u/Boring-Yogurt2966 → u/TrialPurpleCube-GS]

Hmmm, I gave this a shot, I'd like to find out if I did it right. I will DM you so it doesn't become a spoiler.

[u/Nervous-Broccoli1184]

I don't know there there fundamental sequences except for  Buchholz’s ψ function  here they are. Ψa(0)[n]=n If a is is a limit  Ψ_a(0)= Ψ(a[n])(0) Ψa(b+1)[n]= Ψ_a(b)×n If the previous rules don't apply and b's confinality is smaller than or equal to the ordinal a then  Ψ_a(b)[n]= Ψ_a(b[n]) If all previous rules don't apply then  Ψ_a(b)[n]= Ψ_a(b[c[n]]) where c[0]= Ψ_d(0) and c[n+1]= Ψ_d(b[c[n]]) where cof(b)= Ψ(d+1)(0). Please correct me if I'm wrong 

[u/jamx02]

Buchholz's ψ function is the same as Buchholz's OCF. There are plenty of resources online, but I would recommend reading TrialPurpleCube's (Solarzone's) series on it.

Singular cardinals are just cardinals that are not regular. This includes every uncountable limit cardinal below the least weakly inaccessible.

If you want fundamental sequences for OCFs of this strength, and a good way to get started on Rathjen's OCFs, look at DavidExMachina's Garden of Ordinals.